Quotient ring
In , a branch of , a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the s of and the s of . It is a specific example of a , as viewed from the general setting of . R'' and a ''I in R'', and constructs a new ring, the quotient ring , whose elements are the of ''I in R'' subject to special ''+ and ⋅'' operations.}} Quotient rings are distinct from the so-called 'quotient field', or , of an as well as from the more general 'rings of quotients' obtained by . Formal quotient ring construction Given a ring ''R and a two-sided ideal I'' in ''R, we may define an ~ on R'' as follows: :''a ~ b'' is in ''I. Using the ideal properties, it is not difficult to check that ~ is a . In case a'' ~ ''b, we say that a'' and ''b are congruent I''. The of the element ''a in R'' is given by : [''a] = a'' + ''I := { a'' + ''r : r'' in ''I }. This equivalence class is also sometimes written as a'' mod ''I and called the "residue class of a'' modulo ''I". The set of all such equivalence classes is denoted by ; it becomes a ring, the '''factor ring' or quotient ring of R'' modulo ''I, if one defines * (a'' + ''I) + (b'' + ''I) = (a'' + ''b) + I''; * (''a + I'')(''b + I'') = (''a b'') + ''I. (Here one has to check that these definitions are . Compare and .) The zero-element of is , and the multiplicative identity is . The map p'' from ''R to defined by is a , sometimes called the 'natural quotient map' or the 'canonical homomorphism. Examples *The quotient ring } is to R'', and is the {0}, since, by our definition, for any ''r in R'', we have that }} (where {0} is the zero ring), which is isomorphic to R'' itself. This fits with the rule of thumb that the larger the ideal ''I, the smaller the quotient ring . If ''I is a proper ideal of R'', i.e., , then is not the zero ring. * s '''Z and the ideal of s, denoted by 2'Z'. (Odd numbers form a coset.) Then the quotient ring has only two elements (one for each coset), zero for the even numbers and one for the odd numbers;}} applying the definition, }}, where 2'Z' is the ideal of even numbers. It is naturally isomorphic to the with two elements, F'2. Intuitively: if you think of all the even numbers as 0, then every integer is either 0 (if it is even) or 1 (if it is odd and therefore differs from an even number by 1). is essentially arithmetic in the quotient ring (which has n'' elements). * s in the variable ''X with coefficients, and the ideal + 1)}} consisting of all multiples of the polynomial + 1}}. The quotient ring + 1)}} is naturally isomorphic to the field of s C', with the class [''X] playing the role of the i''. The reason is that we "forced" + 1 = 0}}, i.e. = −1}}, which is the defining property of i''.}} *Generalizing the previous example, quotient rings are often used to construct s. Suppose ''K is some and f'' is an in ''K[X'']. Then is a field whose over ''K is f'', which contains ''K as well as an element . *One important instance of the previous example is the construction of the finite fields. Consider for instance the field with three elements. The polynomial + 1}} is irreducible over F'3 (since it has no root), and we can construct the quotient ring . This is a field with elements, denoted by '''F'9. The other finite fields can be constructed in a similar fashion. *The s of are important examples of quotient rings in . As a simple case, consider the real variety x''2 = ''y''3 } }} as a subset of the real plane '''R'2. The ring of real-valued polynomial functions defined on V'' can be identified with the quotient ring − ''Y''3)}}, and this is the coordinate ring of ''V. The variety V'' is now investigated by studying its coordinate ring. *Suppose ''M is a C∞- , and p'' is a point of ''M. Consider the ring of all C∞-functions defined on ''M and let I'' be the ideal in ''R consisting of those functions f'' which are identically zero in some ''U of p'' (where ''U may depend on f''). Then the quotient ring is the ring of s of C∞-functions on ''M at p''. *Consider the ring ''F of finite elements of a *'R'. It consists of all hyperreal numbers differing from a standard real by an infinitesimal amount, or equivalently: of all hyperreal numbers x'' for which a standard integer ''n with exists. The set I'' of all infinitesimal numbers in *'R', together with 0, is an ideal in ''F, and the quotient ring is isomorphic to the real numbers '''R'. The isomorphism is induced by associating to every element x'' of ''F the of x'', i.e. the unique real number that differs from ''x by an infinitesimal. In fact, one obtains the same result, namely R', if one starts with the ring ''F of finite hyperrationals (i.e. ratio of a pair of s), see . Alternative complex planes The quotients , , and are all isomorphic to '''R' and gain little interest at first. But note that )}} is called the plane in geometric algebra. It consists only of linear binomials as "remainders" after reducing an element of R'[''X] by X'' . This alternative complex plane arises as a whenever the algebra contains a and a . Furthermore, the ring quotient − 1)}} does split into and , so this ring is often viewed as the . Nevertheless, an alternative complex number is suggested by j as a root of − 1}}, compared to i as root of + 1 = 0}}. This plane of s normalizes the direct sum by providing a basis }} for 2-space where the identity of the algebra is at unit distance from the zero. With this basis a may be compared to the of the . Quaternions and alternatives Suppose ''X and Y'' are two, non-commuting, s and form the }}. Then Hamilton’s s of 1843 can be cast as : \mathbf{R} \langle X, Y \rangle / ( X^2 + 1, Y^2 + 1, XY + YX ) . If − 1}} is substituted for + 1}}, then one obtains the ring of s. Substituting minus for plus in both the quadratic binomials also results in split-quaternions. The implies that XY has as its square : (XY)(XY) = X''(''YX)Y'' = −''X(XY)Y'' = −''XXYY = −1. The three types of s can also be written as quotients by use of the free algebra with three indeterminates R' and constructing appropriate ideals. Properties Clearly, if R'' is a , then so is ; the converse however is not true in general. The natural quotient map ''p has I'' as its ; since the kernel of every ring homomorphism is a two-sided ideal, we can state that two-sided ideals are precisely the kernels of ring homomorphisms. The intimate relationship between ring homomorphisms, kernels and quotient rings can be summarized as follows: ''the ring homomorphisms defined on are essentially the same as the ring homomorphisms defined on R that vanish (i.e. are zero) on I. More precisely, given a two-sided ideal I'' in ''R and a ring homomorphism whose kernel contains I'', there exists precisely one ring homomorphism with (where ''p is the natural quotient map). The map g'' here is given by the well-defined rule for all ''a in R''. Indeed, this can be used to ''define quotient rings and their natural quotient maps. As a consequence of the above, one obtains the fundamental statement: every ring homomorphism induces a between the quotient ring and the image im(''f). (See also: .) The ideals of R'' and are closely related: the natural quotient map provides a between the two-sided ideals of ''R that contain I'' and the two-sided ideals of (the same is true for left and for right ideals). This relationship between two-sided ideal extends to a relationship between the corresponding quotient rings: if ''M is a two-sided ideal in R'' that contains ''I, and we write for the corresponding ideal in (i.e. ), the quotient rings and are naturally isomorphic via the (well-defined!) mapping . In and , the following statement is often used: If }} is a ring and I'' is a , then the quotient ring is a ; if ''I is only a , then is only an . A number of similar statements relate properties of the ideal ''I to properties of the quotient ring . The states that, if the ideal ''I is the intersection (or equivalently, the product) of pairwise ideals I''1, ..., ''Ik, then the quotient ring is isomorphic to the of the quotient rings , . For algebras over a ring An ''A over a R'' is a ring itself. If ''I is an ideal in A'' (closed under ''R-multiplication), then A'' / ''I inherits the structure of an algebra over R and is the '''quotient algebra. References Category:Advanced mathematics